Decay estimates for the Cauchy problem for the damped extensible beam equation

We study the Cauchy problem for the nonlinear Landau–Ginzburg equation [Formula: see text] where α, β ∈ C with dissipation condition ℜα > 0. We are interested in the subcritical case [Formula: see text]. We assume that θ = | ∫ u0(x) dx| ≠ 0 and ℜδ (α, β) > 0, where [Formula: see text] Furthermore we suppose that the initial data u0 ∈ L1 are such that (1+|x|)au0 ∈ L1, with sufficiently small norm ε = ‖(1 + |x|)a u0 ‖1, where a ∈ (0,1). Also we assume that σ is sufficiently close to [Formula: see text]. Then there exists a unique solution of the Cauchy problem (*) such that [Formula: see text] satisfying the following time decay estimates for large t > 0[Formula: see text] Note that in comparison with the corresponding linear case the decay rate of the solutions of (*) is more rapid.

Download Full-text

Decay Estimates of Solutions to the Cauchy Problem for a Wave Equation with a Bounded Nonlinear Dissipation

Zeitschrift für Analysis und ihre Anwendungen ◽

10.4171/zaa/1350 ◽

2008 ◽

pp. 179-194

Author(s):

Abbès Benaissa ◽

Soufiane Mokeddem

Keyword(s):

Cauchy Problem ◽

Wave Equation ◽

Decay Estimates ◽

Nonlinear Dissipation ◽

Estimates Of Solutions ◽

The Cauchy Problem

Download Full-text

Global well-posedness and temporal decay estimates to the fractional Cahn–Hilliard equation in ℝ N \mathbb{R}^{N}

Forum Mathematicum ◽

10.1515/forum-2018-0288 ◽

2019 ◽

Vol 31 (3) ◽

pp. 803-814

Author(s):

Ning Duan ◽

Xiaopeng Zhao

Keyword(s):

Cauchy Problem ◽

Decay Rate ◽

Weak Solutions ◽

Well Posedness ◽

Decay Estimates ◽

Hilliard Equation ◽

Temporal Decay ◽

The Cauchy Problem ◽

Higher Order Derivative ◽

Cahn Hilliard Equation

AbstractThis paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in{\mathbb{R}^{N}}({N\in\mathbb{N}^{+}}), provided that the initial datum is sufficiently small. In addition, the{L^{p}}-norm ({1\leq p\leq\infty}) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

Download Full-text

On the decay property of solutions to the Cauchy problem of the semilinear beam equation with weak damping for large initial data

Nonlinear Dynamics in Partial Differential Equations ◽

10.2969/aspm/06410507 ◽

2019 ◽

Cited By ~ 1

Author(s):

Hiroshi Takeda ◽

Shuji Yoshikawa

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Decay Property ◽

Beam Equation ◽

Large Initial Data ◽

The Cauchy Problem ◽

Weak Damping

Download Full-text

Decay Rates of The Solution to the Cauchy Problem of the Type III Timoshenko Model Without Any Mechanical Damping

Demonstratio Mathematica ◽

10.1515/dema-2015-0026 ◽

2015 ◽

Vol 48 (3) ◽

Author(s):

Belkacem Said-Houari

Keyword(s):

Cauchy Problem ◽

Heat Conduction ◽

Decay Rates ◽

Order System ◽

Decay Estimates ◽

Pointwise Estimates ◽

Type Iii ◽

The Cauchy Problem ◽

Mechanical Damping ◽

The Stability

AbstractIn this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow and, in the case of nonequal wave speeds, are of regularity-loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term.

Download Full-text